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The Euler-Maclaurin formula helps to relate sums and integrals. I am particularly interested in one case of equation and want to get it clarified. $$\lim_{x\to\infty} \left(\sum_{n=0}^{x}f(n)-\int_{0}^{x}f(u)du\right)$$ What will this lead to according to Euler maclaurin formula? (I have assumed that $f(x)$ is a continuous function and defined for all $x\in\mathbb{R^+}$)

Thanks in advance.

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    $\begingroup$ $\sum_{n=0}^\infty f(n)-\int_0^\infty f(u)du=\tfrac{1}{2}f(0)-\tfrac{1}{12}f'(0)+\tfrac{1}{720}f'''(0)-\tfrac{1}{30240}f'''''(0)+\cdots$ $\endgroup$
    – Carlo Beenakker
    Commented Oct 10, 2021 at 19:24

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